There is a clear need for new statistical tools for accommodating interactions in assessing the health effects of mixtures of chemical exposures. For a review of recent developments, refer to
[1]}.
A broad set of strategies have been taken in this literature. The first extends linear regression to include a quadratic term characterizing pairwise interactions; for recent examples, refer to [2]}, [3]}. This approach can trivially identify synergistic versus antagonistic interactions through the signs of the quadratic coefficients, but relies on a highly restrictive parametric model. An alternative is to rely on nonparametric regression; for example, using Bayesian Additive Regression Trees (BART) [4]} or Gaussian processes (GP) [5]}. Bayesian Kernel Machine Regression (BKMR) [6]} implements GP regression with variable selection motivated by the mixtures problem. These approaches do not identify synergistic or antagonistic interactions, and can lead to overly-wiggly and difficult to interpret dose response surfaces. MixSelect [7]} bridges between parametric and nonparametric approaches via an additive expansion. However, it lacks flexibility to characterize nonlinear main effects and interactions due to its reliance on quadratic regression. To simplify dose response modeling,
[8]} propose profile regression based on clustering exposures, while [9]} use a single index model based on a weighted sum of the exposures. Neither approach allows for inferences on interactions.
